cp's OEIS Frontend

A322833 Squarefree MM-numbers of strict uniform regular multiset multisystems. Squarefree numbers whose prime indices all have the same number of prime factors counted with multiplicity, and such that the product of the same prime indices is a power of a squarefree number.

%I A322833 #5 Dec 28 2018 13:59:50
%S A322833 1,2,3,5,7,11,13,15,17,19,23,29,31,33,41,43,47,51,53,55,59,67,73,79,
%T A322833 83,85,93,97,101,103,109,113,123,127,131,137,139,149,151,155,157,161,
%U A322833 163,165,167,177,179,181,187,191,199,201,205,211,227,233,241,249,255
%N A322833 Squarefree MM-numbers of strict uniform regular multiset multisystems. Squarefree numbers whose prime indices all have the same number of prime factors counted with multiplicity, and such that the product of the same prime indices is a power of a squarefree number.
%C A322833 A multiset multisystem is a finite multiset of finite multisets. A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}.
%C A322833 A multiset multisystem is uniform if all parts have the same size, regular if all vertices appear the same number of times, and strict if there are no repeated parts. For example, {{1,2,2},{1,3,3}} is uniform, regular, and strict, so its MM-number 13969 belongs to the sequence. Note that the parts of parts such as {1,2,2} do not have to be distinct, only the multiset of parts.
%e A322833 The sequence of all strict uniform regular multiset multisystems, together with their MM-numbers, begins:
%e A322833    1: {}           59: {{7}}         157: {{12}}        269: {{2,8}}
%e A322833    2: {{}}         67: {{8}}         161: {{1,1},{2,2}} 271: {{1,10}}
%e A322833    3: {{1}}        73: {{2,4}}       163: {{1,8}}       277: {{17}}
%e A322833    5: {{2}}        79: {{1,5}}       165: {{1},{2},{3}} 283: {{18}}
%e A322833    7: {{1,1}}      83: {{9}}         167: {{2,6}}       293: {{1,11}}
%e A322833   11: {{3}}        85: {{2},{4}}     177: {{1},{7}}     295: {{2},{7}}
%e A322833   13: {{1,2}}      93: {{1},{5}}     179: {{13}}        311: {{1,1,1,1,1,1}}
%e A322833   15: {{1},{2}}    97: {{3,3}}       181: {{1,2,4}}     313: {{3,6}}
%e A322833   17: {{4}}       101: {{1,6}}       187: {{3},{4}}     317: {{1,2,5}}
%e A322833   19: {{1,1,1}}   103: {{2,2,2}}     191: {{14}}        327: {{1},{10}}
%e A322833   23: {{2,2}}     109: {{10}}        199: {{1,9}}       331: {{19}}
%e A322833   29: {{1,3}}     113: {{1,2,3}}     201: {{1},{8}}     335: {{2},{8}}
%e A322833   31: {{5}}       123: {{1},{6}}     205: {{2},{6}}     341: {{3},{5}}
%e A322833   33: {{1},{3}}   127: {{11}}        211: {{15}}        347: {{2,9}}
%e A322833   41: {{6}}       131: {{1,1,1,1,1}} 227: {{4,4}}       349: {{1,3,4}}
%e A322833   43: {{1,4}}     137: {{2,5}}       233: {{2,7}}       353: {{20}}
%e A322833   47: {{2,3}}     139: {{1,7}}       241: {{16}}        367: {{21}}
%e A322833   51: {{1},{4}}   149: {{3,4}}       249: {{1},{9}}     373: {{1,12}}
%e A322833   53: {{1,1,1,1}} 151: {{1,1,2,2}}   255: {{1},{2},{4}} 381: {{1},{11}}
%e A322833   55: {{2},{3}}   155: {{2},{5}}     257: {{3,5}}       389: {{4,5}}
%t A322833 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A322833 Select[Range[100],And[SquareFreeQ[#],SameQ@@PrimeOmega/@primeMS[#],SameQ@@Last/@FactorInteger[Times@@primeMS[#]]]&]
%Y A322833 Cf. A005117, A007016, A112798, A302242, A306017, A319056, A319189, A320324, A321698, A321699, A322554, A322703.
%K A322833 nonn
%O A322833 1,2
%A A322833 _Gus Wiseman_, Dec 27 2018